SWBAT change the parameters of a series of situations, and they will see how these changes affect the graph of a function.

The ability to decide whether or not the value of a parameter is "realistic" in a given context provides a scaffold toward a more general ability to interpret a mathematical result.

5 minutes

Today's opener is pure review, and I consider it optional. Often, students will show up to today's class and want to jump right into their work on the Spending & Saving Project. If that's the vibe, I skip this opener. Few things make me happier than when my students take control like that.

Just in case they need a little warm up, I have this opener ready to go, and it takes us back to arithmetic and geometric sequences. I provide four algebraic rules, and the task is to write the first five terms in each sequence. I'm hoping that students are seeing these rules in a new light, and that they can differentiate fluently between arithmetic and geometric sequences, or linear and exponential functions.

30 minutes

**Ideally, the Classroom Feels Like a Workshop**

As in my last two lessons, today is a work day. We're working on the Spending & Saving Project for the fourth consecutive day. For some background, please take a look at the previous three lessons:

- Day 1: The introduction to the project, including "Part 1a: Saving Money"
- Day 2: I introduce "Part 1b: Value Up, Value Down"
- Day 3 (yesterday): I introduce "Part 2" of the project, where students are creating tables and graphs for each situation from Part 1

For the last two days, I've tried to maximize the amount of class time kids have to work on their projects. As they finish each part of the project, I provide them with the next. Today, it's the same deal. For anyone who needs time getting caught up or getting help, they've got it. For anyone who is ready, I provide the next part of the project.

**Next Up is Part 2: Changing the Parameters**

On Part 2: Changing the Parameters, students will continue to work with the graphs they've made already. Here's a video narrative in which I describe this task.

The first thing students have to do on this part of the project is come up with numbers that make sense to them - a reasonable cost for a new computer, a reasonable amount of money to save each month, etc. Because they're making their own common-sense decisions about how to adjust the parameters in each situation, and because of the work they've already done to get to this point, students can work pretty autonomously on this part of the project. The best way to get a feeling for this part of the project is to look at some student work, which I've included here.

- Three complete graphs for the first situation will look like this. Students change the original cost of item they'd like to save up for, which is the y-intercept of a linear function. Note that the x-intercept changes too. I try to make sure that students can tell me what it means when the x-intercept moves to the left or right - it represents the changing amount of time that it will take to save for more or less expensive items. The weekly savings amount does not change, so all three lines have the same slope.
- Here is what the completed Changing Parameters handout should look like. This is a companion to the graphs Students are asked to write a little bit about how the graphs change, and to answer a few more interpretation questions.
- Three complete graph for the the second situation will look like this. Here, the monthly savings amount changes, and that's a rate, so the slope of each line is different. This time, the starting amount, or y-intercept, is the same for each line.
- Revision will certainly be necessary on this part of the project. It's important to work with students to help them see revision as a vital step in a process, and not a penalty for doing something wrong.

There are few tips that I'll often provide, if necessary. As they work on the new versions of their linear functions, I remind students that it's not necessary to plot every single point. Two points are enough to sketch the line, and plotting three or four (I usually emphasize the intercept(s), plus two more) is enough to get a good graph. In some cases, I'll help students choose new parameters that will make it a bit easier to fit their "changed" functions on their original graph. We'll take a look at how they've scaled their axes, and rewrite the parameters accordingly.

A lot of conversations I'll have with kids are based on checking in on what is a "realistic" value for each parameter is a key step in helping kids interpret the meaning of a parameter in an algebraic rule.

In the next day or so (probably not during today's lesson), I'll project this handout on the screen and use it to give a little overview of what's happening on this part of the project. This is an opportunity for me to define what "parameters" are for the whole class at once. Ideally, this is old news to almost everyone by the time I share it here.

3 minutes

With a few minutes left, I close today's lesson just like I did yesterday. I applaud students on the work they've done today, and I might give a few shout outs to individual students who have worked particularly hard, helped others, or gained new insights.

Then, I tell students to take a moment to gather their work, assess what they've done so far, and decide what they need to work on tonight. "You all have homework tonight!" I say. "It's different for each of you, because you're each at a different point in the project, but I expect that everyone can finish whatever you're currently working on tonight." Sometimes, I give a specific time suggestion, like, "Everyone should do 30 minutes of math homework tonight." I find that suggesting a *maximum* amount of work can make homework seem less daunting to kids. "Sure, it's hard to do piles and piles of homework, for hours upon hours, but what if you commit to working for 30 minutes tonight, and see how much you can accomplish?"